C₁₃×#12971 #15699310

order	3328
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₃×#12971

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃	B(4,𝔽₂)
commutator	C₂×Q₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×Q₈	C₂³×C₁₃
prime radical	C₂³×C₁₃	#120
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂³×C₁₃ ◁ C₂²×Q₈×C₁₃ ◁ G

424 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 49
transitive degree	μ_t(G) = 832

A minimal faithful permutation representation — 3 generators on 49 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,17,16)(18,19)(20,30)(21,27)(22,35)(23,38)(24,29)(25,31)(26,34)(28,40)(32,48)(33,45)(36,43)(37,44)(39,41)(42,49)(46,47);(14,16,17,15)(18,20)(19,25)(21,31)(22,36)(23,39)(24,33)(26,41)(27,45)(28,42)(29,43)(30,35)(32,40)(34,49)(37,47)(38,46)(44,48);(18,21,24,22)(19,26,29,44)(20,31,33,36)(23,37,40,34)(25,41,43,48)(27,28,35,38)(30,46,45,42)(32,49,39,47)

Conjugacy classes & character table

442 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 5959);